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arXiv:hep-th/0010164 · Non-perturbative Quantum Effects

Universal ratios of the renormalization group

Giuseppe Mussardo

Published Sep 25, 2000• 1 citation
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Abstract

The scaling form of the free-energy near a critical point allows for the definition of various universal ratios of thermodynamical amplitudes.Together with the critical exponents they characterize the universality classes and may be useful experimental quantities.We show how these universal quantities can be computed for a particular class of universality by using several Quantum Field Theory methods

DOI
10.22323/1.006.0047

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The scaling form of the free-energy near a critical point allows for the definition of various universal ratios of thermodynamical amplitudes.Together with the critical exponents they characterize the universality classes and may be useful experimental quantities.We show how these universal quantities can be computed for a particular class of universality by using several Quantum Field Theory methods
The scaling form of the free energy near a critical point allows for the definition of various thermodynamical amplitudes and the determination of their dependence on the microscopic nonuniversal scales. Universal quantities can be obtained by considering special combinations of the amplitudes. Together with the critical exponents they characterize the universality classes and may be useful quantities for their experimental identification. We compute the universal amplitude ratios for the tricritical Ising model in two dimensions by using several theoretical methods from perturbed conformal field theory and scattering integrable quantum field theory. The theoretical approaches are further supported and integrated by results coming from a numerical determination of the energy eigenvalues and eigenvectors of off-critical systems in an infinite cylinder.
Writing thermodynamic quantities near a critical point as ${f}_{i}\ensuremath{\sim}{A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, we show that to leading order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ one has $\frac{{a}_{i}}{{a}_{j}}=\frac{({\ensuremath{\lambda}}_{i}\ensuremath{-}{{\ensuremath{\lambda}}_{i}}^{0})}{({\ensuremath{\lambda}}_{j}\ensuremath{-}{{\ensuremath{\lambda}}_{j}}^{0})}$, where ${{\ensuremath{\lambda}}_{i}}^{0}$ is the mean-field value of ${\ensuremath{\lambda}}_{i}$. This ratio is also equal to $\frac{({\ensuremath{\lambda}}_{i, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{i})}{({\ensuremath{\lambda}}_{j, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{j})}$, where ${\ensuremath{\lambda}}_{i, \mathrm{eff}}$ is derived from a fit of data with ${f}_{i}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i, \mathrm{eff}}}$. Various experiments are analyzed and compared to these predictions.
Critical phenomena, characterized by abrupt changes in the macroscopic properties of a system, exhibit a profound degree of universality, meaning that vastly different physical systems can display identical behavior near their critical points. This universality is typically understood through the lens of scaling laws and the Renormalization Group theory, which classify critical points into distinct universality classes. This paper proposes and explores the "Entropic Law of Critical Universality," an overarching principle positing that not only do thermodynamic quantities scale universally, but the fundamental information-theoretic measures of a system, particularly entropy in its various forms (thermodynamic entropy, entanglement entropy, Renyi entropy), also conform to universal scaling behaviors that uniquely characterize these universality classes. We review how entanglement entropy exhibits universal logarithmic corrections at quantum critical points, linked to the central charge of underlying conformal field theories. We also discuss how specific entropy behaves non-monotonically near quantum chromodynamics critical points and how entropic measures can identify phase transitions and their classes in various condensed matter systems, including graphene and spin models. The paper synthesizes theoretical frameworks and empirical evidence to demonstrate that entropy serves as a powerful and often more direct probe of the universal properties underlying critical phenomena, offering a comprehensive perspective on the shared information landscape of diverse critical systems.
In the last few years the derivative expansion of the nonperturbative renormalization group has proven to be a very efficient tool for the precise computation of critical quantities. In particular, recent progress in the understanding of its convergence properties allowed for an estimate of the error bars as well as the precise computation of many critical quantities. In this work we extend previous studies to the computation of several universal amplitude ratios for the critical regime of O(N) models using the derivative expansion of the nonperturbative renormalization group at order O(∂^{4}) for three-dimensional systems.
Precision calculation of universal amplitude ratios in O(N)O(N)O(N) universality classes: Derivative Expansion results at order O(∂4)\mathcal{O}(\partial^4)O(∂4)
In the last few years the derivative expansion of the Non-Perturbative Renormalization Group has proven to be a very efficient tool for the precise computation of critical quantities. In particular, recent progress in the understanding of its convergence properties allowed for an estimate of the error bars as well as the precise computation of many critical quantities. In this work we extend previous studies to the computation of several universal amplitude ratios for the critical regime of O(N)O(N)O(N) models using the derivative expansion of the Non-Perturbative Renormalization Group at order O(∂4)\mathcal{O}(\partial^4)O(∂4) for three dimensional systems.